= Exponential-linear separable potential
{title2=$H(x)=\sum_i(a_i e^{x_i}-c_i x_i),\quad a_ic_i>0$}
Let $a_i,c_i$ be nonzero real constants with $a_ic_i>0$. The real <function> $H:\mathbb R^n\to\mathbb R$ displayed in the title has exactly one <stationary point>:
$$
x_i^*=\log(c_i/a_i),\qquad
H(x^*)=\sum_i\bigl[c_i-c_i\log(c_i/a_i)\bigr].
$$
Indeed the <partial derivatives> are $a_i e^{x_i}-c_i$, each of which has precisely one zero. The <Hessian matrix> at that point is $\operatorname{diag}(c_1,\ldots,c_n)$.
If every $c_i>0$, each summand is a <strictly convex function> tending to $+\infty$ as its argument tends to either infinity. Consequently $H$ has a unique <global minimum>. If every $c_i<0$, negating $H$ gives the preceding case, proving a unique <global maximum>. If both signs occur, restricting to two coordinate directions of opposite signs gives a <saddle point of a scalar function>. No degenerate <Hessian matrix> is possible under the nonzero assumptions.
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